$\begin{aligned} & \text { If }|\vec{a}|=\sqrt{26},|\vec{b}|=7 \\ & |\vec{a} \times \vec{b}|=35 \text {, find } \vec{a} \cdot \vec{b}\end{aligned}$

Vector identity relates cross and dot products: $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$

Given $|\vec{a}| = \sqrt{26}$, $|\vec{b}| = 7$, $|\vec{a} \times \vec{b}| = 35$:

$35^2 + (\vec{a} \cdot \vec{b})^2 = (\sqrt{26} \cdot 7)^2$

$1225 + (\vec{a} \cdot \vec{b})^2 = 49 \times 26 = 1274$

$(\vec{a} \cdot \vec{b})^2 = 1274 - 1225 = 49$

$\vec{a} \cdot \vec{b} = \pm 7$

Choose the positive value consistent with the options: $\vec{a} \cdot \vec{b} = 7$

Final answer: $\boxed{7}$